Separation of Coadjoint Orbits of Generalized Diamond Lie Groups

Let $G$ be a type I connected and simply connected generalized diamond Lie group defined as the semidirect product of a $d$-dimensional Abelian Lie group $N$ with $(2n+1)$-dimensional Heisenberg Lie group $\mathbb{H}_{2n+1}$ for some $(n,d)\in(\mathbb{N}^*)^2$. Let $\mathfrak{g}^*/G$ denote the set of coadjoint orbits of $G$, where $\mathfrak{g}^*$ is the dual vector space of the Lie algebra $\mathfrak{g}$ of $G$. In this paper, we address the problem of separation of coadjoint orbits of $G$. We first specify the setting where $d=1$; we prove that the closed convex hull of coadjoint orbit $\mathcal{O}$ in $\mathfrak{g}^*$ does characterize $\mathcal{O}$. Whenever $d\ge2$, we provide a separating overgroup $G^+$ of $G$. More precisely, we extend the group $G$ to an overgroup denoted by $G^+$, containing $G$ as a subgroup, and we give an injective map $\varphi$ from $\mathfrak{g}^*$ into $(\mathfrak{g}^+)^*$, the dual vector space of Lie algebra $\mathfrak{g}^+$ of $G^+$ sending each $G$-orbit in $\mathfrak{g}^*$ to the $G^+$-orbit in $(\mathfrak{g}^+)^*$ in such a manner that the closed convex hull of $\varphi(\mathcal{O})$ does characterize $\mathcal{O}$, where $\mathcal{O}$ is a $G$-orbit in $\mathfrak{g}^*$.